Linear Algebra and Geometry

Preface CHAPTER 1 Linear Spaces and Linear Mappings 1 Linear Spaces 2 Basis and Dimension 3 Linear Mappings 4 Matrices 5 Subspaces and Direct Sums 6 Quotient Spaces 7 Duality 8 The Structure of a Linear Mapping 9 The Jordan Normal Form 10 Normed Linear Spaces 11 Functions of Linear Operators 12 Complexification and Decomplexification 13 The Language of Categories 14 The Categorical Properties of Linear Spaces CHAPTER 2 Geometry of Spaces with an Inner Product 1 On Geometry 2 Inner Products 3 Classification Theorems 4 The Orthogonalization Algorithm and Orthogonal Polynomials 5 Euclidean Spaces 6 Unitary Spaces 7 Orthogonal and Unitary Operators 8 Self-Adjoint Operators 9 Self-Adjoint Operators in Quantum Mechanics 10 The Geometry of Quadratic Forms and the Eigenvalues of Self-Adjoint Operators 111 Three-Dimensional Euclidean Space 12 Minkowski Space 13 Symplectic Space 14 Witt's Theorem and Witt's Group 15 Clifford Algebras CHAPTER 3 Affine and Projective Geometry 1 Affine Spaces, Affine Mappings, and Affine Coordinates 2 Affine Groups 3 Affine Subspaces 4 Convex Polyhedra and Linear Programming 5 Affine Quadratic Functions and Quadrics 6 Projective Spaces 7 Projective Duality and Projective Quadrics 8 Projective Groups and Projections 9 Desargues' and Pappus' Configurations and Classical Projective Geometry 10 The Kahler Metric 11 Algebraic Varieties and Hilbert Polynomials CHAPTER 4 Multilinear Algebra 1 Tensor Products of Linear Spaces 2 Canonical Isomorphisms and Linear Mappings of Tensor Products 3 The Tensor Algebra of a Linear Space 4 Classical Notation 5 Symmetric Tensors 6 Skew-Symmetric Tensors and the Exterior Algebra of a Linear Space 7 Exterior Forms 8 Tensor Fields 9 Tensor Products in Quantum Mechanics